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A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem

  • * Corresponding author: Wei Wang

    * Corresponding author: Wei Wang

The second author is supported by NSFC grant 12025101, 11431001

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  • In this paper, we prove a symmetric property for the indices for symplectic paths in the enhanced common index jump theorem (cf. Theorem 3.5 in [6]). As an application of this property, we prove that on every compact Finsler manifold $ (M, \, F) $ with reversibility $ \lambda $ and flag curvature $ K $ satisfying $ \left(\frac{\lambda}{\lambda+1}\right)^2<K\le 1 $, there exist two elliptic closed geodesics whose linearized Poincaré map has an eigenvalue of the form $ e^{\sqrt {-1}\theta} $ with $ \frac{\theta}{\pi}\notin{\bf Q} $ provided the number of closed geodesics on $ M $ is finite.

    Mathematics Subject Classification: Primary: 53C22, 53C60; Secondary: 58E10.


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